Consider the following gamble (where you pay me a dollar):
What are the expected returns? Variance?
\[ \begin{aligned} E(r_{\text{gamble}}) &= 0.5 \times (2-1) + 0.5 \times (0.5 - 1) = 0.25\\ \sigma^{2}(r_{\text{gamble}}) &= 0.5 \times (1 - 0.25)^2 + 0.5 \times (-0.5-0.25)^2 = 0.5625 \\ \sigma(r_{\text{gamble}}) &= \sqrt{ 0.5625 } = 0.75 \end{aligned} \]
Would you take it? What if you paid me 1.25?
At what cost would you become unwilling to take this deal?
\[ U(r) = E(r) - \frac{1}{2} \times A \times \sigma^{2}(r) \] * \(A>0\) implies risk aversion * \(A = 0\) implies risk neutrality * \(A < 0\) implies risk seeking
U(r) provides an ordering over investments, given their returns and risks
This particular function can be derived under a number of different assumptions and is widely used by practitioners and academics
\[ U(r) = E(r) - \frac{1}{2} \times A \times \sigma^{2}(r) \]
For \(A = 1\):
\[ U(r) = E(r) - \frac{1}{2} \times A \times \sigma^{2}(r) \]
For \(A = 0.5\):
\[ U(r) = E(r) - \frac{1}{2} \times A \times \sigma^{2}(r) \]
For \(A = 0.0\):
\[ U(r) = E(r) - \frac{1}{2} \times A \times \sigma^{2}(r) \]
| Risk Aversion | A=0.04 | A=0.5 | A=0.78 | A=1 |
|---|---|---|---|---|
| Risk Free | 0.03 | 0.03 | 0.03 | 0.03 |
| Risky | 0.24 | 0.11 | 0.03 | -0.03 |
For risk aversion of 0.5, do you prefer \(r_{f}\) or investment \(r_{p}\)?
Suppose the constant of risk aversion is 0.78?
Now what about if you can choose to invest in both assets, but with different weights?
What weight (\(w\)) should you allocate to the risky asset?
What would we expect to see in reality?
\[ \begin{aligned} E(r_{\text{blended}}) & = E\big(wr_{p} + (1-w)r_{f}\big)\\ & = wE(r_{p}) + (1-w)E(r_{f})\\ & = r_{f} + wE(r_{p}-r_{f}) \end{aligned} \]
\[ \begin{aligned} \sigma^{2}(r_{\text{blended}}) &= \text{Var}\big(wr_{p} + (1-w)r_{f}\big)\\ &= w^{2}\sigma^{2}(r_{p}) + (1-w)^{2}\sigma^{2}(r_{f}) + 2w(1-w)\sigma(r_{p}, r_{f})\\ &= w^{2}\sigma^{2}(r_{p})\\ \sigma(r_{\text{blended}})&= w \sigma(r_{p}) \end{aligned} \]
\[ E(r_{\text{blended}}) = r_{f} + \sigma(r_{\text{blended}})\frac{E(r_{p} - r_{f})}{\sigma_{p}} \]
\[ E(r_{\text{blended}}) = r_{f} + \sigma(r_{\text{blended}})\frac{E(r_{p} - r_{f})}{\sigma_{p}} \]
\[ E(r_{\text{blended}}) = r_{f} + \sigma(r_{\text{blended}})\frac{E(r_{p} - r_{f})}{\sigma_{p}} = 0.03 + \sigma(r_{\text{blended}}) \frac{0.22}{0.75} \]
The capital allocation line shows all risk‐return combinations available based on choice of w.
The slope of the capital allocation line (the Sharpe Ratio) prices the risk-return tradeoff
In our example, the Sharpe Ratio is 0.29 (0.22/0.75)
\[ E(r_{\text{blended}}) = r_{f} + \sigma(r_{\text{blended}})\frac{E(r_{p} - r_{f})}{\sigma_{p}} \]
\[ \max_{w} U(r_{blend}) = r_{f} + w E(r_{p} - r_{f}) - \frac{1}{2} A w^{2} \sigma^{2}(r_{p}) \]
\[ w^{*} = \frac{E(r_{p} - r_{f})}{A\sigma^{2}(r_{p})} \]
| A | \(w^{*}\) | \(E(r_{\text{blended}})\) | \(\sigma_{\text{blended}}\) |
|---|---|---|---|
| 0.25 | 1.56 | 0.37 | 1.17 |
| 0.5 | 0.78 | 0.2 | 0.51 |
| 0.78 | 0.49 | 0.14 | 0.37 |
| 1 | 0.39 | 0.12 | 0.29 |
Now suppose instead of choosing a mix between a risky and a risk‐free asset, we have two risky assets (but no risk free asset).
The expected return for the portfolio of risky assets A and B is \[ E(r_{p}) = w_{a}E(r_{a}) + (1-w)E(r_{b}) \] where \(w_{a}\) is the weight on stock A.
The variance and standard deviation of the portfolio are:
\[ \begin{aligned} \sigma^{2}(r_{\text{p}}) &= \text{Var}\big(w_{a}r_{a} + (1-w_{a})r_{b}\big)\\ &= w_{a}^{2}\sigma^{2}(r_{a}) + (1-w)^{2}\sigma^{2}(r_{b}) + 2w_{a}(1-w_{a})\sigma(r_{a}, r_{b})\\ &= w_{a}^{2}\sigma^{2}(r_{a}) + (1-w)^{2}\sigma^{2}(r_{b}) + 2w_{a}(1-w_{a})\rho_{a,b}\sigma(r_{a})\sigma(r_{b}) \end{aligned} \]
| Asset | \(E(r)\) | \(\sigma(r)\) |
|---|---|---|
| A | 0.25 | 0.75 |
| B | 0.1 | 0.25 |
| Asset | \(E(r)\) | \(\sigma(r)\) |
|---|---|---|
| A | 0.25 | 0.75 |
| B | 0.1 | 0.25 |
\[\rho_{a,b} = 1\]
| Asset | \(E(r)\) | \(\sigma(r)\) |
|---|---|---|
| A | 0.25 | 0.75 |
| B | 0.1 | 0.25 |
\[\rho_{a,b} = -1\]
| Asset | \(E(r)\) | \(\sigma(r)\) |
|---|---|---|
| A | 0.25 | 0.75 |
| B | 0.1 | 0.25 |
\[\rho_{a,b} = 0\]
Formally, we can derive it using calculus, as \[ \begin{aligned} \max_{w_{a}} U(r_{p}) & = E(r_{p}) - \frac{1}{2} A \sigma^{2}(r_{p})\\ \text{s.t.} \; E(r_{p}) &= E(r_{p}) = w_{a}E(r_{a}) + (1-w)E(r_{b})\\ \text{and} \; \sigma^{2}(r_{\text{p}}) &= w_{a}^{2}\sigma^{2}(r_{a}) + (1-w)^{2}\sigma^{2}(r_{b}) + 2w_{a}(1-w_{a})\rho_{a,b}\sigma(r_{a})\sigma(r_{b}) \end{aligned} \]
Solving directly (by plugging in and taking derivative w.r.t. \(w_{a}\)):
\[ \begin{aligned} w_{a}^{*} &= \frac{E(r_{a}) - E(r_{b})}{A(\sigma_{a}^{2} + \sigma_{b}^{2} - 2\rho_{a,b}\sigma_{a}\sigma_{b})} + \frac{\sigma_{b}^{2} - \rho_{a,b}\sigma_{a}\sigma_{b}}{\sigma_{a}^{2} + \sigma_{b}^{2} - 2\rho_{a,b}\sigma_{a}\sigma_{b}}\\ &=\frac{E(r_{a}) - E(r_{b}) + A \big(\sigma_{b}^{2} - \rho_{a,b}\sigma_{a}\sigma_{b}\big)}{A(\sigma_{a}^{2} + \sigma_{b}^{2} - 2\rho_{a,b}\sigma_{a}\sigma_{b})} \end{aligned} \]
Add back our riskless asset
What is our Capital Allocation Line when we combine either A or B with our riskless asset?
To illustrate more clearly the point, let \(\sigma(r_{A})=0.5\):
| Asset | \(E(r)\) | \(\sigma(r)\) |
|---|---|---|
| A | 0.25 | 0.50 |
| B | 0.1 | 0.25 |
We can pick a different combination of assets A and B from the frontier and combine it with our riskless asset.
This “tangency portfolio” is the mean-variance efficient (MVE) portfolio.
\[ SR = \frac{E(r_{p} - r_{f})}{\sigma_{p}} \]
\[ \begin{aligned} \max_{w_{a}} & \frac{E(r_{p} - r_{f})}{\sigma_{p}}\\ \text{s.t.} \; & E(r_{p}) = w_{a}E(r_{a}) + (1-w_{a})E(r_{b})\\ \text{and} \; & \sigma_{p}^{2} = w_{a}^{2}\sigma^{2}(r_{a}) + (1-w)^{2}\sigma^{2}(r_{b}) + 2w_{a}(1-w_{a})\rho_{a,b}\sigma(r_{a})\sigma(r_{b}) \end{aligned} \]
\[ w_{a,MVE}^{*} = \frac{E(r_{a} - r_{f})\sigma^{2}_{b}- E(r_{b} - r_{f})\sigma_{a}\sigma_{b}\rho_{a,b}}{E(r_{a}-r_{f})\sigma_{b}^{2} + E(r_{b}-r_{f})\sigma_{a}^{2} - \big[E(r_{a}-r_{f}) + E(r_{b}-r_{f})\big]\sigma_{a}\sigma_{b}\rho_{a,b}} \]
Key thing to notice – this optimal portfolio does not include anything related to investor risk aversion (\(A\))
Takeaway: All investors do best by choosing the same risky portfolio and then deciding how much to allocate to the riskless asset based on individual preferences
Now we have a simple two step recipe for an optimal portfolio, based on our taste for risk!
Let’s do an example with the following stocks: Ford, IBM, Microsoft, Netflix and Walmart.
Take their monthly returns from 2010-2018. What’s their average monthly return and standard deviation? What about the Sharpe Ratio?
|
Ticker |
Monthly Return |
Monthly SD |
Sharpe Ratio |
|---|---|---|---|
|
F |
0.006 |
0.075 |
0.086 |
|
IBM |
0.004 |
0.047 |
0.083 |
|
MSFT |
0.016 |
0.063 |
0.250 |
|
NFLX |
0.054 |
0.180 |
0.300 |
So why does diversification shift the frontier?
https://learn-investments.rice-business.org/portfolios/three-assets
So why does diversification shift the frontier?
| F | IBM | MSFT | NFLX | WMT |
|---|---|---|---|---|
| 1.000 | 0.279 | 0.357 | 0.210 | 0.159 |
| 0.279 | 1.000 | 0.339 | 0.107 | 0.221 |
| 0.357 | 0.339 | 1.000 | 0.216 | 0.135 |
| 0.210 | 0.107 | 0.216 | 1.000 | -0.066 |
| 0.159 | 0.221 | 0.135 | -0.066 | 1.000 |
\[ \begin{aligned} \min_{w_{a}} \sigma^{2}_{p} &= w_{a}^{2}\sigma^{2}(r_{a}) + (1-w_{a})^{2}\sigma^{2}(r_{b}) + 2w_{a}(1-w_{a})\rho_{a,b}\sigma(r_{a})\sigma(r_{b})\\ w_{a, MVP}^{*} &= \frac{\sigma_{b}^{2} - \sigma_{a}\sigma_{b}\rho_{a,b}}{\sigma_{a}^{2} + \sigma_{b}^{2} - 2\sigma_{a}\sigma_{b}\rho_{a,b}} \end{aligned} \]
\[ \begin{aligned} \min_{\mathbf{w}} \sigma^{2}_{p} &= \mathbf{w}' \Sigma \mathbf{w}\\ \text{s.t.} \; \mathbf{w}'\mathbf{1} &= 1\\ \end{aligned} \]
\[ \begin{aligned} \hat{\mu} &= \frac{s_{1}\sigma_{2}^{2} + s_{2}\sigma_{1}^{2}}{\sigma_{2}^{2} + \sigma_{1}^{2}}\\ &= \frac{\frac{1}{\sigma_{1}^{2}} s_{1} + \frac{1}{\sigma_{2}^{2}}s_{2}}{\frac{1}{\sigma_{1}^{2}} + \frac{1}{\sigma_{2}^{2}}}\\ \end{aligned} \]
\[ \begin{aligned} \hat{\mu}_{JS} &= (1- \hat{w})\bar{X} + \hat{w} \mu \mathbf{1}\\ \mu &= \frac{\mathbf{1}'\Sigma^{-1}\bar{X}}{\mathbf{1}'\Sigma^{-1}\mathbf{1}}\\ \hat{w} &= \frac{N+2}{(N+2) + (X - \mu\mathbf{1})'T\Sigma^{-1}(X - \mu\mathbf{1})} \end{aligned} \]
Taken to an extreme, this gives us some naive portfolios
For example, what happens if I put all my weight on the MVP prior (e.g. returns are flat across all stocks)
Note the MVP (a “naive” portfolio) can be motivated by Bayesian methods
Consider the out-of-sample performance of different models in various combinations of assets
What if we constrain our portfolio weights?
Simple example: only long positions (to minimize losses)
Mean-variance analysis works off the assumption that all investors care about is the mean and variance of returns
This delivers some powerful takeaways
What’s wrong with this?